2.3-bo'lim : Chiziqli tenglamalarni qo'llash
5. Nasos hovuzni 7 soatda, boshqa nasos esa bir xil hovuzni 12 soatda bo‘shatishi mumkin. Hovuzni bo'shatish uchun ikkala nasos birga ishlaganda qancha vaqt ketadi?
Yechim
Shunday qilib, agar hovuzni bo'shatishni bitta ish deb hisoblasak, hovuzni bo'shatish uchun ishlaydigan ikkala nasosni tavsiflovchi quyidagi so'z tenglamasiga ega bo'lamiz.
\[\left( \begin{array}{c}{\mbox{Portion of job done}}\\ {\mbox{ by first pump}}\end{array} \right) + \left( \begin{array}{c}{\mbox{Portion of job done}}\\ {\mbox{ by second pump}}\end{array} \right) = 1{\mbox{ Job}}\]Biz bilamizki, Ish qismi = Ish tezligi X ish vaqti, shuning uchun bu quyidagi so'z tenglamasini beradi.
\[\left( \begin{array}{c}{\mbox{Work Rate of}}\\ {\mbox{ first pump}}\end{array} \right)\left( \begin{array}{c}{\mbox{ Work Time}}\\ {\mbox{of first pump}}\end{array} \right) + \left( \begin{array}{c}{\mbox{Work Rate of}}\\ {\mbox{ second pump}}\end{array} \right)\left( \begin{array}{c}{\mbox{ Work Time}}\\ {\mbox{of second pump}}\end{array} \right) = 1\]2-qadam
Bizga har bir nasosning ish stavkalari kerak bo'ladi va buning uchun biz muammo bayonotida mavjud bo'lgan ma'lumotlardan har bir nasosning alohida ishlashi va ishni alohida bajaradigan har bir nasos uchun quyidagi so'z tenglamasidan foydalanishimiz mumkin.
\[\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of pump}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{ of pump}}\end{array} \right) = 1\]Bizda bor birinchi nasos uchun,
\[\left( \begin{array}{c}{\mbox{ Work Rate}}\\ {\mbox{of first pump}}\end{array} \right)\left( 7 \right) = 1\hspace{0.25in} \Rightarrow \hspace{0.25in}{\mbox{Work Rate of first pump = }}\frac{1}{7}\]va ikkinchi nasos uchun bizda,
\[\left( \begin{array}{c}{\mbox{ Work Rate}}\\ {\mbox{of second pump}}\end{array} \right)\left( {12} \right) = 1\hspace{0.25in} \Rightarrow \hspace{0.25in} {\mbox{Work Rate of second pump = }}\frac{1}{{12}}\]3-qadam
Endi \(t\) hovuzni bo'shatish uchun ikkala nasosning birgalikda ishlashiga ketadigan vaqt miqdori bo'lsin. Bu va biz ikkinchi bosqichda topilgan ish stavkalaridan foydalanib, birinchi bosqichdagi so'z tenglamamiz shunday bo'ladi:
\[\begin{align*}\left( {\frac{1}{7}} \right)\left( t \right) + \left( {\frac{1}{{12}}} \right)t & = 1\\ \frac{{19}}{{84}}t & = 1\end{align*}\]4-qadam
Endi biz buni \(t\) uchun hal qila olamiz.
\[\frac{{19}}{{84}}t = 1\hspace{0.25in} \Rightarrow \hspace{0.25in}t = \frac{{84}}{{19}} = 4.4211\]Shunday qilib, ikkala nasos ham birgalikda ishlasa, hovuzni bo'shatish uchun taxminan 4,4211 soat vaqt ketadi.